Riot, why haven't you done this yet?

I'd be interested in this. Maybe a dev will comment.

Or just remove boots from the game and give everyone the bonus movespeed on some kind of trinket enchant

cause selling your boots for an another item is a comment to a set strategy.

There ahve been a few lcs games where some people never bought boots the whole game. This is an unsual style but it does have its uses.

Also with triforce and zepher you can make up for the boot speed on certain melee champs.

These are the reasons why boots do not get there own slot. Cause while 99% of the games will have boots on everyone. Someone is going to rock your world and not get them at all. They will have more damage and or AS. they are willing to risk not getting away and just brawling with you every time.

There are a few champs like Kassadin who can be built without boots. So, that's a consideration.

Mostly, though, I'd be curious to see how late game would be rebalanced. Imagine a Ryze with RoA, Seraph's, and Void Staff, but also Spirit Visage, Frozen Heart and GA. Or a six-defensive item Mundo. Also, even if every laner is farming equally, the change would widen the disparity between laners and junglers/supports late game, which could have dubious consequences.

It's an interesting idea, at least.

Would you also be making boots free, like trinkets? If you're making boots free like trinkets, then balance is incredibly difficult, since some boots are worth a lot more than others in terms of stats they offer. Honestly I think it's because boots offer non-negligible stats to a champion, whereas trinkets only affect vision (yes they can help certain champions in other ways, such as ward jumping, but for the most part, they aren't as vital to any specific champion). For example, Hecarim benefits a great deal from bonus move speed, where a lvl 6+ Kassadin could probably do just fine without the move speed.

Honestly, I don't see why there needs to be a boot specific slot; this argument is basically the same as adding an extra item slot. I can see where you're coming from where mobility, like vision, are basically universally sought after in champions, but vision was something that Riot wanted to have greater emphasis in the game, where no one really needs encouragement to prioritize mobility. Not to mention, mobility is a stat of the champion, not a factor outside of the champion that affect the game like vision does. Honestly I think it would make more sense for them to add an extra ward slot, rather than a boot slot (though I don't think they should do this). The argument seems to be the same as why Riot doesn't give each team another ban in Draft pick; it's to force people to do more with less.

Making the trinket slot for boots might not be a bad idea. Upgrade the trinket when you want a higher move speed. The problem with that is it would make the hypercarries even harder to play against.. Could you imagine Master Yi with a full build and moving at 700 ms.

Or Katarina and her ult doing 1k damage to 4 people

I agree that movespeed is something that really needs to be worked on, but giving boots a slot of there own might be a bit much.

Having a boot slot either forces every champ to have boots and only 5 other item slots, or creates an additional slot for only boots and increases late game from 5 items to 6 items +boots. This would mean scaling champions would be even more ridiculous if it got to that point.